Partial Differential Equations

1 The single first-order equation.- 1. Introduction.- 2. Examples.- 3. Analytic Solution and Approximation Methods in a Simple Example.- Problems.- 4. Quasi-linear Equations.- 5. The Cauchy Problem for the Quasi-linear Equation.- 6. Examples.- Problems.- 7. The General First-order Equation for a Function of Two Variables.- 8. The Cauchy Problem.- 9. Solutions Generated as Envelopes.- Problems.- 2 Second-order equations: hyperbolic equations for functions of two independent variables.- 1. Characteristics for Linear and Quasi-linear Second-order Equations.- 2. Propagation of Singularities.- 3. The Linear Second-order Equation.- Problems.- 4. The One-Dimensional Wave Equation.- Problems.- 5. Systems of First-order Equations (Courant-Lax Theory).- 6. A Quasi-linear System and Simple Waves.- Problem.- 3 Characteristic manifolds and the Cauchy problem.- 1. Notation of Laurent Schwartz.- Problems.- 2. The Cauchy Problem.- Problems.- 3. Cauchy-Kowalewski Theorem.- Problems.- 4. The Lagrange-Green Identity.- 5. The Uniqueness Theorem of Holmgren.- 6. Distribution Solutions.- Problems.- 4 The Laplace equation.- 1. Green's Identity, Fundamental Solutions.- Problems.- 2. The Maximum Principle.- Problems.- 3. The Dirichlet Problem, Green's Function, and Poisson's Formula.- Problems.- 4. Proof of Existence of Solutions for the Dirichlet Problem Using Subharmonic Functions ("Perron's Method").- Problem.- 5. Solution of the Dirichlet Problem by Hilbert-Space Methods.- Problems.- 5 Hyperbolic equations in higher dimensions.- 1. The Wave Equation in n-dimensional Space.- (a) The method of spherical means.- Problems.- (b) Hadamard's method of descent.- Problems.- (c) Duhamel's principle and the general Cauchy problem.- Problem.- (d) Initial-boundary-value problems ("Mixed" problems).- Problems.- 2. Higher-order Hyperbolic Equations with Constant Coefficients.- (a) Standard form of the initial-value problem.- Problem.- (b) Solution by Fourier transformation.- Problems.- (c) Solution of a mixed problem by Fourier transformation.- (d) The method of plane waves.- Problems.- 3. Symmetric Hyperbolic Systems.- (a) The basic energy inequality.- Problems.- (b) Existence of solutions by the method of finite differences.- Problems.- 6 Higher-order elliptic equations with constant coefficients.- 1. The Fundamental Solution for Odd n.- Problems.- 2. The Dirichlet Problem.- Problems.- 7 Parabolic equations.- 1. The Heat Equation.- (a) The initial-value problem.- Problems.- (b) Maximum principle, uniqueness, and regularity.- Problems.- (c) A mixed problem.- Problems.- 2. The Initial-value Problem for General Second-order Linear Parabolic Equations.- (a) The method of finite differences and the maximum principle.- (b) Existence of solutions of the initial-value problem.- Problems.